Determine how many solutions exist for the system of equations. ${-9x+3y = -30}$ ${y = 3x-10}$
Solution: Convert both equations to slope-intercept form: ${-9x+3y = -30}$ $-9x{+9x} + 3y = -30{+9x}$ $3y = -30+9x$ $y = -10+3x$ ${y = 3x-10}$ ${y = 3x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-10}$ ${y = 3x-10}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${-9x+3y = -30}$ is also a solution of ${y = 3x-10}$, there are infinitely many solutions.